Strongly Stabilizing Controller Synthesis for a ... - Semantic Scholar
Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008
Strongly Stabilizing Controller Synthesis for a Class of MIMO Plants ⋆ ∗ ¨ H. Ozbay A. N. G¨ unde¸s ∗∗ ∗ Department of Electrical & Electronics Engineering, Bilkent University, Ankara, 06800 Turkey (e-mail: [email protected]) ∗∗ Department of Electrical & Computer Engineering, University of California, Davis, CA 95616, USA (e-mail: [email protected])
Abstract: The strong stabilization problem (i.e., stabilization by a stable feedback controller) is studied for unstable MIMO plants with arbitrary number of finitely many poles but no more than two blocking zeros in the extended right half plane. Simple strongly stabilizing controllers, of order not exceeding that of the plant, are obtained for such plants satisfying the parity interlacing property. Connections with earlier design methods are illustrated: for this particular class of plants, it is shown that a sufficient condition appearing in earlier publications is equivalent to the parity interlacing property and hence it is also necessary for the existence of strongly stabilizing controllers. The results are illustrated with numerical examples. Keywords: Feedback Stabilization; Stable Controllers; Unstable MIMO Plants. 1. INTRODUCTION In this paper we discuss the strong stabilization problem, i.e., feedback stabilization using a stable controller, for a class of linear time-invariant (LTI), multi-input multi-output (MIMO) systems. It is well known that a given plant is strongly stabilizable if and only if it satisfies the parity interlacing property (p.i.p), that is, the number of poles between every pair of real blocking zeros in the extended right half plane is even, Youla et al. (1974); Vidyasagar (1985). In the single-input multi-output (SISO) case, an interpolation-based procedure is available to derive strongly stabilizing controllers for a given plant, see Doyle et al. (1992); Vidyasagar (1985). Moreover, for such plants, a parameterization of all strongly stabilizing controllers can be obtained using interpolation with infinite dimensional transfer functions, Vidyasagar (1985). On the other hand, extensions of these interpolation techniques to MIMO plants are not currently available. Nevertheless, a vast literature exists on strong stabilization of MIMO plants, see e.g. Campos-Delgado and Zhou (2003); Choi and Chung (2001); Chou et al. (2007); Gumussoy and Ozbay (2005); Halevi (1994); Lee and Soh (2002); Petersen (2006), and their references. In these papers, strongly stabilizing controllers are obtained under certain sufficient conditions. In addition to strong stabilization, most of these papers also consider an H∞ or H2 like performance conditions. In this work, rather than investigating new sufficient conditions for the construction of strongly stabilizing controllers for general MIMO plants, we derive a simple design procedure for a restricted class of plants satisfying the PIP (i.e., the design procedure works for all strongly stabilizable plants in this class). The plants we consider satisfy the PIP ⋆ This work is supported in part by TUB ¨ ITAK ˙ under grant no. EEEAG-105E065.
978-1-1234-7890-2/08/$20.00 © 2008 IFAC
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and have at most two real blocking zeros in the extended right half plane. In this approach, there is no restriction on the number and location of the poles, and on the number of left half plane zeros. Furthermore, the controller order does not exceed that of the plant. We also show that the sufficient condition of Zeren and Ozbay (2000) for general MIMO plants is also necessary for the class of plants considered here. In Section 2 we define the class of MIMO plants considered for strong stabilization. The design procedure is given in Section 3. Connections with the approach of Zeren and Ozbay (2000) are illustrated in Section 4 with an example. Concluding remarks are made in Section 5. The notation used in the paper is fairly standard, in particular R+e = {z ∈ R | z ≥ 0} ∪ {∞} represents the (extended) non-negative real axis, and define C+e := C+ ∪ {∞}. The set of real rational functions that are bounded and analytic in C+ is denoted by RH∞ . We also say that a matrix valued function is in RH∞ if all its entries are in RH∞ . The norm symbol, k · k, used in the paper is the usual operator norm defined in RH∞ . 2. PROBLEM DEFINITION Consider the standard feedback control system shown in Figure 1, where P is the plant to be controlled and C is the controller to be designed. It is assumed that all entries of the matrices P and C are proper rational functions, and that P and C have no unstable hidden-modes. Let P = Dp−1 Np be a left-coprime-factorization (LCF) of the plant and let C = Nc Dc−1 be a right-coprime-factorization (RCF) of the controller, where Dp , Np , Nc , Dc are appropriate size matrices with all entries in RH∞ , and Dp−1 , Dc−1 are proper. A controller C = Nc Dc−1 stabilizes the feedback system if and only if the matrix U = Dp Dc + N p N c (1)
10.3182/20080706-5-KR-1001.0263
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
Case ℓ = 0: First consider the trivial case, where ℓ = 0, i.e., Npz = I and hence, P = Dp−1 Npr has no blocking-zeros in C+e . Then for any arbitrary unimodular matrix Uc , the † stable controller C = Npr (Uc − Dp ) strongly stabilizes P since (1) is satisfied with Dc = I, Nc = C, which leads to U = Uc . Fig. 1. Feedback Control System is unimodular, i.e., all entries of U and U −1 are in RH∞ . Moreover, the stabilizing controller is itself stable if and only if this condition is satisfied with a unimodular Dc ; in this case we say that C strongly stabilizes P . Let z1 , . . . , zℓ ∈ R+e be the non-negative real-axis blockingzeros of P in the extended right-half-plane C+e , i.e., Np (zk ) = 0 for k = 1, . . . , ℓ. Then P satisfies the PIP if and only if det[Dp (zk )] is sign invariant for k = 1, . . . , ℓ (see e.g. Vidyasagar (1985)). The plants under consideration for strongly stabilizing controller synthesis here are full row-rank and have no restrictions on the location or number of their poles. The left-half-plane zeros are also unrestricted. These plants may have at most two non-negative real-axis blockingzeros (including infinity) and no other transmission-zeros in C+e . More specifically, the numerator Np in any LCF P = Dp−1 Np can be factored as Np = Npz Npr , where Npz contains the blocking-zeros in C+e , and Npr is right † invertible by a transfer matrix Npr , whose entries are in ∞ RH (since P has no other-transmission zeros in C+e ); then Npz can be expressed as Npz =
ℓ∞ Y
ℓ−ℓ Y∞ (s − zk ) 1 I , (s + ai ) (s + bk ) i=1
(2)
k=1
where ai > 0, bk > 0 are positive real for 1 ≤ i ≤ ℓ∞ , † 1 ≤ k ≤ ℓ − ℓ∞ . The controller C = Npr Ncz Dc−1 stabilizes −1 P = Dp Npz Npr if and only if U = Dp Dc + Npz Ncz
(3)
is unimodular, and C strongly stabilizes P if in addition Dc is unimodular in (3). The goal then is to find appropriate Ncz , Dc whose entries in RH∞ , with Dc being unimodular, such that U is unimodular. Although design for the general MIMO case is done under certain sufficient conditions as in Barabanov (1996); Campos-Delgado and Zhou (2003); Chou et al. (2003, 2007); Petersen (2006) and Zeren and Ozbay (1999), the design procedure developed here works for every plant satisfying the PIP in the class considered here, with 0 ≤ ℓ∞ ≤ ℓ ≤ 2. 3. STRONGLY STABILIZING CONTROLLER DESIGN Let the plant be P = Dp−1 Npz Npr , with Npz as in (2). We assume that Dp (s) is non-unimodular; otherwise P is stable and hence the trivial controller C = 0 strongly stabilizes it. Another point to note is that the order of the controllers to be derived in this section depend on the particular coprime factorization which is non-unique. Therefore, in order to obtain the lowest order controller one should try to find the lowest order Dp and Npr .
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Case ℓ∞ = ℓ = 1: In this case (2) becomes 1 Npz = I , a>0, (4) s+a 1 Dp−1 Npr has a blocking-zero at infinity. and hence, P = s+a In the SISO case, these plants have relative-degree one and are called minimum-phase. The PIP is automatically satisfied since there are no finite non-negative zeros. Then the stable controller † C = (s + a)Npr (s) ( Dp (∞) − Dp (s) ) (5) strongly stabilizes P since (1) is satisfied with unimodular Dc = Dp (∞)−1 , Nc = CDp (∞)−1 , which leads to U = I. Note that Dp (∞)−Dp (s) is strictly proper, hence the term (s + a)(Dp (∞) − Dp (s)) is proper. Case ℓ∞ = 0 < ℓ = 1: In this case (2) becomes s−z Npz = I , z≥0, b>0, (6) s+b −1 and hence, P (s) = (s−z) (s+b) Dp Npr has one finite blockingzero at z ≥ 0. In the SISO case, these plants have relativedegree zero. The PIP is automatically satisfied because there is only one zero in C+e . Then the stable controller (s + b) † C= N (s) ( Dp (z) − Dp (s) ) (7) (s − z) pr strongly stabilizes P since (1) is satisfied with unimodular Dc = Dp (z)−1 , Nc = CDp (z)−1 , which leads to U = I. Case ℓ∞ = ℓ = 2: In this case (2) becomes 1 Npz = I , a1 > 0 , a 2 > 0 , (8) (s + a1 )(s + a2 ) 1 and hence, P = (s+a1 )(s+a Dp−1 Npr has two blocking2) zeros at infinity. In the SISO case, these plants have relative-degree two and are called minimum-phase. The PIP is automatically satisfied since there are no finite nonnegative zeros. Define D∞ (s) := (Dp (s)Dp (∞)−1 − I) which is strictly proper and stable. Choose any positive real β > 0 satisfying β > k s D∞ (s) k . (9) Let the controller be (s + a1 )(s + a2 ) † Npr (s) (Dp (∞) − Dp (s)). (10) C=β (s + β) Since Dp (∞) − Dp (s) is strictly-proper, the term (s+a1 )(s+a2 ) (Dp (∞) − Dp (s)) is proper; then all entries of (s+β) C are in RH∞ . With Dc = I, Nc = C, (1) becomes U = Dp (s) +
β (Dp (∞) − Dp (s)) (s + β)
1 sD∞ (s))Dp (∞). s+β For β satisfying (9), by the small-gain theorem, 1 k s+β sD∞ (s)k < 1, which implies U is unimodular. Therefore, the stable controller in (10) strongly stabilizes P . = (I +
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
Case ℓ∞ = 1 < ℓ = 2: In this case (2) becomes (s − z) Npz = I , z ≥ 0 , a > 0, b > 0 , (s + a)(s + b)
(11)
(s−z) Dp−1 Npr has a finite blockingand hence, P = (s+a)(s+b) zero at z ≥ 0 and one at infinity. In the SISO case, these plants have relative-degree one. Define W := Dp (z)−1 Dp (∞), which is obviously nonsingular. For these plants, the PIP is satisfied if and only if det W > 0 since det Dp is sign invariant at the blocking-zeros of P . The strongly stabilizing controller design proposed here assumes a more conservative condition on W : Let all eigenvalues of W = Dp (z)−1 Dp (∞) have positive real parts (which is equivalent to PIP for single-output plants since W > 0). Choose any real β > 0 satisfying (9) as in the case of 2-zeros at infinity above. With Npz as in (11), let the controller be −1 † Npr (s)(I − Dp (s)Dp (z)−1 )Dp (∞)(sI + βW )−1 , C = βNpz (12) (s+b) where (s−z) (I − Dp (s)Dp (z)−1 ) is stable. By assumption, −W has eigenvalues in the open left-half-plane and β > 0 implies (s + a)(sI + βW )−1 is also stable; then all entries of C are in RH∞ . With Dc = I, Nc = C, (1) becomes
U = Dp (s) + β(I − Dp (s)Dp (z)−1 )Dp (∞)(sI + βW )−1 = (sDp (s) + βDp (∞))(sI + βW )−1 1 sD∞ (s)) Dp (∞)(s + β)(sI + βW )−1 = (I + s+β 1 By (9) the term (I + s+β sD∞ (s)) is unimodular. Since Dp (∞) is nonsingular, and (s + β)(sI + βW )−1 is unimodular, the controller (12) is strongly stabilizing P .
The strongly stabilizing controller in (12) is designed under the condition that W = Dp (z)−1 Dp (∞) has eigenvalues in the open C+ ; while this is a sufficient condition for PIP for MIMO plants, it is equivalent to PIP for single-output systems. Case ℓ∞ = 0 < ℓ = 2: We only consider the case where (at least) one of the two finite blocking-zeros is at zero. In this case (2) becomes s (s − z) Npz = I , z ≥ 0 , b1 > 0 , b2 > 0 , (13) (s + b1 )(s + b2 ) s(s−z) Dp−1 Npr has a finite blockingand hence, P = (s+b 1 )(s+b2 ) zero at s = 0 and another at z ≥ 0. In the SISO case, these plants have relative-degree zero. Define V := Dp (0)−1 Dp (z), which is obviously nonsingular. For these plants, the PIP is satisfied if and only if det V > 0 since det Dp is sign invariant at the blocking-zeros of P . Similar to the previous case, the strongly stabilizing controller design proposed here assumes a more conservative condition on V : Let all eigenvalues of V = Dp (0)−1 Dp (z) have positive real parts (which is equivalent to PIP for single-output plants since V > 0). Let us define the stable transfer matrix Dp (s)Dp (0)−1 − I D0 (s) := s and choose any positive real ρ > 0 satisfying ρ k D0 (s) k < 1 . (14)
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Let the controller be (s + b1 )(s + b2 ) † C= Npr (s) [ Dp (z) − Dp (s) ] (sI + ρV )−1 , (s − z) (15) (s+b1 ) where (s−z) [ Dp (z1 ) − Dp (s) ] is stable. By assumption, −V has eigenvalues in the open left-half-plane and ρ > 0 implies (s + b2 )(sI + ρV )−1 is also stable; then all entries of C are in RH∞ . With Dc = I, Nc = C, (1) becomes U = Dp (s) + s(Dp (z) − Dp (s)) (sI + ρV )−1 = ( Dp (s)ρV + sDp (z) ) (sI + ρV )−1 = ((s + ρ − ρ)I + ρDp (s)Dp (0)−1 ) Dp (z)(sI + ρV )−1 s = (I + ρ D0 (s)) (s + ρ)Dp (z)(sI + ρV )−1 s+ρ s By (14) the term (I + s+ρ ρ D0 (s)) is unimodular. Since Dp (z) is nonsingular and (s + ρ)(sI + ρV )−1 unimodular the controller (15) is strongly stabilizing P . 4. CONNECTIONS WITH THE CONDITION OF ¨ ZEREN AND OZBAY In this section we show that a sufficient condition used to design strongly stabilizing controllers in Zeren and Ozbay (2000) is also necessary for the single-output version of the plants considered here. By (1), the plant P = Dp−1 Npz Npr is strongly stabilizable † if and only if there exists stable C = Npr Q, Q ∈ RH∞ , such that U := Dp + Npz Q is unimodular. Define Rp := Dp − I, then U = I + Rp + Npz Q . Using the small-gain theorem, a strongly stabilizing controller can be found if there exists Q ∈ RH∞ such that kRp + Npz Qk < 1. Since Npz (z) = 0 at all blocking-zeros z ∈ C+e , of P , a feasible Q ∈ RH∞ can be found only if kRp (z)k < 1. Now consider the plants described in Section 3 and let P have a single-output. Therefore Dp ∈ RH∞ and Npz ∈ RH∞ are scalars anr Npr has a single row. The only C+e zeros of the plant are in R+e , i.e., none are complex. By (2), a coprime factorization is P = Dp−1 Np = Dp−1 Npz Npr , where Npz contains the C+e zeros and Npr has a singlecolumn right-inverse in RH∞ . Without loss of generality, the denominator Dp ∈ RH∞ of P = Dp−1 Np can be 1 ⊂ R+ , p i ≥ expressed as follows: Let Pr = {pi }ki=1 0 denote the non-negative real poles and Pi = {ei ± 2 ⊂ C+ \ R+ , ei ≥ 0 denote the complex poles of jfi }ki=1 P in C+ . Then decompose Dp (s) as Dp (s) = Dpr (s)Dpc (s) (16) where k1 Y
Dpr (s) =
(s − pi ) (s + p i + ci ) i=1
Dpc (s) =
k2 Y
( (s − ei )2 + fi2 ) ( (s + ei )2 + fi2 + di s ) i=1
for some ci , di > 0 . Since Dp (∞) = 1, Rp = Dp − I is strictly-proper. Suppose that the plant is strongly stabilizable, equivalently the PIP holds (since Dp (∞) > 0, PIP implies
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
Dp (z) > 0). Suppose that the only C+e zeros of P are in R+e , i.e., P has no complex zeros in C+e . Let zmax ≤ ∞ ∈ R+e denote the largest zero of P . Suppose that pi ≤ zmax for all pi ∈ Pr , which always holds if P is strictly-proper. We now show that under these assumptions, |R(z)| < 1 at all R+e zeros of P : Let 0 ≤ z ≤ zmax be any of the R+e zeros of the plant, i.e., Npz (z) = 0. It is obvious that Q 2 ( (z−ei )2 +fi2 ) Dpc (z) = ki=1 satisfies 0 < Dpc (z) ≤ 1. ( (z+ei )2 +fi2 +di z ) For Dpr (z), there are two possibilities: Either this zero z > pi for all pi ∈ Pr , or z < pi for an even number of the poles pi ∈ Pr since the PIP implies an even number of positive real poles of P in the interval [z, zmax] (as well as between any two consecutive positive real zeros). In Qk1 (z−pi ) both cases, it is obvious that Dpr (z) = i=1 (z+pi +ci ) also satisfies 0 < Dpr (z) ≤ 1. Therefore, 0 < Dpr (z)Dpc (z) ≤ 1 implies |Rp (z)| = |1 − Dp (z)| < 1 at all R+e zeros of P . In the single-output case, the strongly stabilizing controller † can be written as C = Q = Npr Qp for some Qp ∈ RH∞ . Then using the sufficient condition of Zeren and Ozbay (2000), there exists a strongly stabilizing controller if inf ∞ kRp + Npz Qp k∞ < 1. (17) Qp ∈RH
For the single-output plant classes (4) and (8), condition 1 (17) is satisfied if and only if |Rp (∞)| < 1: For Npz = s+a , the optimal solution is Qp = (s + a)(Rp (∞) − Rp (s)) = (s + a)(Dp (∞) − Dp (s)); † † i.e., Q = Npr Qp is the same as controller C = Npr Ncz 1 in (5). For Npz = (s+a1 )(s+a2 ) , since Dp (∞) = 1 and β satisfies (9), the optimal solution is −1 −1 βNpz βNpz Qp = (Rp (∞) − Rp (s)) = (Dp (∞) − Dp (s)); s+β s+β † † i.e., Q = Npr Qp is the same as controller C = Npr Ncz in (10). For the cases (6) and (11) condition (17) is satisfied if and only if |Rp (z)| < 1. Necessity is obvious, for the sufficiency see e.g. Section 2.11 of the book by Foias et al (1996). For the case (6), Npz (s) = s−z s+b , the optimal solution is −1 −1 Qp = Npz (Rp (z) − Rp (s)) = Npz (Dp (z) − Dp (s)), † † i.e., Q = Npr Qp is the same as controller C = Npr Ncz in (7). In the case (13) the problem (17) is solvable if and only if |Rp (z)| < 1 and |Rp (0)| < 1. For systems with more than one zero in open C+ solvability condition for (17) is more complicated than just checking |Rp (zi )| < 1; but if one zero is in R+ and the other one is on the boundary of C+e , like in (13), then this statement is correct, see Section 2.11.3 of the book by Foias et al (1996).
We showed above that for (4), (8), (6) and (11) strong stabilizability implies |Rp (z)| < 1, which in turn implies condition (17). Therefore condition (17) is necessary and sufficient for these plants; i.e. there exist strongly stabilizing controllers (equivalently the PIP is satisfied) if and only if there is exists a stable Qp such that kRp + Npz Qp k < 1. Note that, here we do not require that the plant has no poles on the jω-axis, which is an assumption in Zeren and Ozbay (2000). Example. Consider the plant (18), which is taken from Zeren and Ozbay (2000),
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(s + 1)(s − 2) (s + 2)(s − 2) (18) (s2 + 4s + 5)(s − α) (s2 + 4s + 5)(s − α) with α ≥ 0. We see that this plant admits a factorization in the form P = Dp−1 Npz Npr , where P =
s−α s+1 s−2 Npz (s) = (s + a)(s + b) (s + a)(s + b) Npr (s) = 2 1 s + 4s + 5 Dp (s) =
s+2 . s+1
This fits into the form (11). Clearly Npz (2) = 0 and Npz (∞) = 0, so PIP is satisfied if and only if α < 2. The controller proposed in Section 3 for this type of systems is in the form (12), where we can take 1−c 2 s + 4s + 5 † (s + 1) Npr (s) = (s + a)(s + b) c (s + 2)
with any c ∈ R. We have W = Dp (2)−1 Dp (∞) = 3/(2−α), and W > 0 if and only if α < 2 (equivalent to PIP). Now we need to choose β > 0 such that β > ks D∞ (s)k. For this example, ks D∞ (s)k = (α + 1). Applying the formula (12) we obtain 1−c 2 −β(1 + α)(s + 4s + 5) (s + 1) , C= (19) 3β ) c (s + 2) (2 − α)(s + 1)(s + 2−α with any c ∈ R and any β > (α + 1). In this case we have U= =
s−α s+1
−
β(1 + α) (s − 2) 3β (2 − α) (s + 1)(s + (2−α) )
s2 + (β − α)s + β . 3β ) (s + 1)(s + (2−α)
Since β > α and α < 2, U is unimodular. Note that our condition β > ks D∞ (s)k gave β > (α + 1). But in fact β > α is sufficient. This gap is due to the use of the small gain theorem in our design of Section 3 in guaranteeing that U is unimodular. For the same system, we can obtain an alternative controller using the method of Zeren and Ozbay (2000). In that approach first note that (α + 1) Rp (s) = Dp (s) − 1 = − s+1 α+1 and |Rp (2)| = 3 < 1 if and only if α < 2 (equivalent to PIP). Solution of the problem (17) gives (α + 1)(s + b)(s + a) Qp (s) = − 3(s + 1)(1 + εs) for some sufficiently small ε > 0. In order to understand how small ε should be let us now examine U for the † controller C = Npr Qp , U (s) = =
s−α (α + 1)(s − 2) − s+1 3(s + 1)(1 + εs) 2(α+1) − εα εs2 + ( 2−α 3 − εα)s + 3 . (s + 1)(1 + εs)
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
This means that ε satisfying 0 < ε < 2−α 3α gives a strongly stabilizing controller 1−c (α + 1)(s2 + 4s + 5) (s + 1) . C(s) = − (20) c 3(s + 1)(1 + εs) (s + 2) Note that the two controllers (19) and (20) are identical if we set ε = 2−α 3β . 5. CONCLUSIONS In this paper we considered a class of MIMO plants with at most two blocking zeros in Re (no other blocking zeros in C+e ), and no restrictions were imposed on the number and locations of the poles. When they satisfy the PIP, these plants are shown to admit strongly stabilizing controllers and we have constructed such controllers explicitly. The order of the strongly stabilizing controllers developed here is equal to the order of the plant. We have also shown that for the single-output version of the class of plants considered here the sufficient condition of Zeren and Ozbay (2000), given for the construction of strongly stabilizing controllers, is equivalent to PIP. Strongly stabilizing controller design for MIMO plants with higher number of C+ blocking zeros is more difficult. For such plants one may have to check the existing sufficient conditions and design methods proposed earlier, see e.g. Campos-Delgado and Zhou (2003); Choi and Chung (2001); Chou et al. (2003, 2007); Gumussoy and Ozbay (2005); Lee and Soh (2002); Petersen (2006); Zeren and Ozbay (1999, 2000), and their references. For example, when we have two complex conjugate zeros placed in such a way that the PIP is about to be violated (as the imaginary part goes to zero) many of the existing finite dimensional (order of the controller is a few multiples of the plant order) controller design techniques fail. Because in this case the minimum order of the strongly stabilizing controllers can be very large (grows as the imaginary part gets smaller), see Smith and Sondergeld (1986). This is the main reason why we have not considered the two finite zero case. Our goal was to derive simple strongly stabilizing controllers for a class of plants using the PIP only. No additional conditions are imposed for this class of plants. REFERENCES Barabanov, A. E., “Design of H∞ optimal stable controller,” Proc. IEEE Conference on Decision and Control, pp. 734–738, 1996. Campos-Delgado, D. U., and K. Zhou, “A parametric optimization approach to H∞ and H2 strong stabilization,” Automatica, vol. 39, no. 7, pp. 1205–1211, 2003. Choi, Y., and W.K. Chung, “On the stable H∞ controller parameterization under sufficient condition,” IEEE Transactions on Automatic Control, vol.46, pp. 1618–1623, 2001. Chou, Y. S., T. Z. Wu, and J. L. Leu, “On strong stabilization and H∞ strong stabilization problems,” Proc. IEEE Conference on Decision and Control, pp. 5155–5160, 2003. Chou Y. S., J. L. Leu, and Y. C. Chu, “Stable Controller Design for MIMO Systems: An LMI Approach,” IET Control Theory Appl., Vol. 1, pp. 817–829, 2007.
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Doyle, J. C., B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory, Macmillan, NY, 1992. ¨ Foias, C., H. Ozbay, A. Tannenbaum, Robust Control of Infinite Dimensional Systems: Frequency Domain Methods, Lecture Notes in Control and Information Sciences, No. 209, Springer-Verlag, London, 1996. ¨ G¨ um¨ u¸ssoy, S., and H. Ozbay, “Remarks on strong stabilization and stable H∞ controller design,” IEEE Transactions on Automatic Control, vol. 50, pp. 2083–2087, 2005. Halevi, Y., “Stable LQG controllers,” IEEE Transactions on Automatic Control, vol. 39, pp. 2104–2106, 1994. Lee, P. H., and Y. C. Soh, “Synthesis of stable H∞ controller via the chain scattering framework,” Systems and Control Letters, vol.46, pp. 1968–1972, 2002. Petersen, I.,“Robust H∞ control of an uncertain system via a strict bounded real output feedback controller,” Proc. 45th IEEE Conference on Decision and Control, pp. 571–577, 2006. Smith, M. C., and K. P. Sondergeld, “On the order of stable compensators,” Automatica, vol. 22, pp. 127–129, 1986. Vidyasagar, M., Control System Synthesis: A Factorization Approach, Cambridge, MA: MIT Press, 1985. Youla, D. C., J. J. Bongiorno, and C. N. Lu, “Single-loop feedback stabilization of linear multivariable dynamical plants,” Automatica, vol.10, pp. 159–173, 1974. ¨ Zeren, M., and H. Ozbay, “On the synthesis of stable H∞ controllers,” IEEE Transactions on Automatic Control, vol.44, pp. 431–435, 1999. ¨ Zeren, M., and H. Ozbay, “On the strong stabilization and stable H∞ -controller design problems for MIMO systems,” Automatica, vol.36, pp. 1675–1684, 2000.
Strongly Stabilizing Controller Synthesis for a Class of MIMO Plants ⋆ ∗ ¨ H. Ozbay A. N. G¨ unde¸s ∗∗ ∗ Department of Electrical & Electronics Engineering, Bilkent University, Ankara, 06800 Turkey (e-mail: [email protected]) ∗∗ Department of Electrical & Computer Engineering, University of California, Davis, CA 95616, USA (e-mail: [email protected])
Abstract: The strong stabilization problem (i.e., stabilization by a stable feedback controller) is studied for unstable MIMO plants with arbitrary number of finitely many poles but no more than two blocking zeros in the extended right half plane. Simple strongly stabilizing controllers, of order not exceeding that of the plant, are obtained for such plants satisfying the parity interlacing property. Connections with earlier design methods are illustrated: for this particular class of plants, it is shown that a sufficient condition appearing in earlier publications is equivalent to the parity interlacing property and hence it is also necessary for the existence of strongly stabilizing controllers. The results are illustrated with numerical examples. Keywords: Feedback Stabilization; Stable Controllers; Unstable MIMO Plants. 1. INTRODUCTION In this paper we discuss the strong stabilization problem, i.e., feedback stabilization using a stable controller, for a class of linear time-invariant (LTI), multi-input multi-output (MIMO) systems. It is well known that a given plant is strongly stabilizable if and only if it satisfies the parity interlacing property (p.i.p), that is, the number of poles between every pair of real blocking zeros in the extended right half plane is even, Youla et al. (1974); Vidyasagar (1985). In the single-input multi-output (SISO) case, an interpolation-based procedure is available to derive strongly stabilizing controllers for a given plant, see Doyle et al. (1992); Vidyasagar (1985). Moreover, for such plants, a parameterization of all strongly stabilizing controllers can be obtained using interpolation with infinite dimensional transfer functions, Vidyasagar (1985). On the other hand, extensions of these interpolation techniques to MIMO plants are not currently available. Nevertheless, a vast literature exists on strong stabilization of MIMO plants, see e.g. Campos-Delgado and Zhou (2003); Choi and Chung (2001); Chou et al. (2007); Gumussoy and Ozbay (2005); Halevi (1994); Lee and Soh (2002); Petersen (2006), and their references. In these papers, strongly stabilizing controllers are obtained under certain sufficient conditions. In addition to strong stabilization, most of these papers also consider an H∞ or H2 like performance conditions. In this work, rather than investigating new sufficient conditions for the construction of strongly stabilizing controllers for general MIMO plants, we derive a simple design procedure for a restricted class of plants satisfying the PIP (i.e., the design procedure works for all strongly stabilizable plants in this class). The plants we consider satisfy the PIP ⋆ This work is supported in part by TUB ¨ ITAK ˙ under grant no. EEEAG-105E065.
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and have at most two real blocking zeros in the extended right half plane. In this approach, there is no restriction on the number and location of the poles, and on the number of left half plane zeros. Furthermore, the controller order does not exceed that of the plant. We also show that the sufficient condition of Zeren and Ozbay (2000) for general MIMO plants is also necessary for the class of plants considered here. In Section 2 we define the class of MIMO plants considered for strong stabilization. The design procedure is given in Section 3. Connections with the approach of Zeren and Ozbay (2000) are illustrated in Section 4 with an example. Concluding remarks are made in Section 5. The notation used in the paper is fairly standard, in particular R+e = {z ∈ R | z ≥ 0} ∪ {∞} represents the (extended) non-negative real axis, and define C+e := C+ ∪ {∞}. The set of real rational functions that are bounded and analytic in C+ is denoted by RH∞ . We also say that a matrix valued function is in RH∞ if all its entries are in RH∞ . The norm symbol, k · k, used in the paper is the usual operator norm defined in RH∞ . 2. PROBLEM DEFINITION Consider the standard feedback control system shown in Figure 1, where P is the plant to be controlled and C is the controller to be designed. It is assumed that all entries of the matrices P and C are proper rational functions, and that P and C have no unstable hidden-modes. Let P = Dp−1 Np be a left-coprime-factorization (LCF) of the plant and let C = Nc Dc−1 be a right-coprime-factorization (RCF) of the controller, where Dp , Np , Nc , Dc are appropriate size matrices with all entries in RH∞ , and Dp−1 , Dc−1 are proper. A controller C = Nc Dc−1 stabilizes the feedback system if and only if the matrix U = Dp Dc + N p N c (1)
10.3182/20080706-5-KR-1001.0263
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
Case ℓ = 0: First consider the trivial case, where ℓ = 0, i.e., Npz = I and hence, P = Dp−1 Npr has no blocking-zeros in C+e . Then for any arbitrary unimodular matrix Uc , the † stable controller C = Npr (Uc − Dp ) strongly stabilizes P since (1) is satisfied with Dc = I, Nc = C, which leads to U = Uc . Fig. 1. Feedback Control System is unimodular, i.e., all entries of U and U −1 are in RH∞ . Moreover, the stabilizing controller is itself stable if and only if this condition is satisfied with a unimodular Dc ; in this case we say that C strongly stabilizes P . Let z1 , . . . , zℓ ∈ R+e be the non-negative real-axis blockingzeros of P in the extended right-half-plane C+e , i.e., Np (zk ) = 0 for k = 1, . . . , ℓ. Then P satisfies the PIP if and only if det[Dp (zk )] is sign invariant for k = 1, . . . , ℓ (see e.g. Vidyasagar (1985)). The plants under consideration for strongly stabilizing controller synthesis here are full row-rank and have no restrictions on the location or number of their poles. The left-half-plane zeros are also unrestricted. These plants may have at most two non-negative real-axis blockingzeros (including infinity) and no other transmission-zeros in C+e . More specifically, the numerator Np in any LCF P = Dp−1 Np can be factored as Np = Npz Npr , where Npz contains the blocking-zeros in C+e , and Npr is right † invertible by a transfer matrix Npr , whose entries are in ∞ RH (since P has no other-transmission zeros in C+e ); then Npz can be expressed as Npz =
ℓ∞ Y
ℓ−ℓ Y∞ (s − zk ) 1 I , (s + ai ) (s + bk ) i=1
(2)
k=1
where ai > 0, bk > 0 are positive real for 1 ≤ i ≤ ℓ∞ , † 1 ≤ k ≤ ℓ − ℓ∞ . The controller C = Npr Ncz Dc−1 stabilizes −1 P = Dp Npz Npr if and only if U = Dp Dc + Npz Ncz
(3)
is unimodular, and C strongly stabilizes P if in addition Dc is unimodular in (3). The goal then is to find appropriate Ncz , Dc whose entries in RH∞ , with Dc being unimodular, such that U is unimodular. Although design for the general MIMO case is done under certain sufficient conditions as in Barabanov (1996); Campos-Delgado and Zhou (2003); Chou et al. (2003, 2007); Petersen (2006) and Zeren and Ozbay (1999), the design procedure developed here works for every plant satisfying the PIP in the class considered here, with 0 ≤ ℓ∞ ≤ ℓ ≤ 2. 3. STRONGLY STABILIZING CONTROLLER DESIGN Let the plant be P = Dp−1 Npz Npr , with Npz as in (2). We assume that Dp (s) is non-unimodular; otherwise P is stable and hence the trivial controller C = 0 strongly stabilizes it. Another point to note is that the order of the controllers to be derived in this section depend on the particular coprime factorization which is non-unique. Therefore, in order to obtain the lowest order controller one should try to find the lowest order Dp and Npr .
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Case ℓ∞ = ℓ = 1: In this case (2) becomes 1 Npz = I , a>0, (4) s+a 1 Dp−1 Npr has a blocking-zero at infinity. and hence, P = s+a In the SISO case, these plants have relative-degree one and are called minimum-phase. The PIP is automatically satisfied since there are no finite non-negative zeros. Then the stable controller † C = (s + a)Npr (s) ( Dp (∞) − Dp (s) ) (5) strongly stabilizes P since (1) is satisfied with unimodular Dc = Dp (∞)−1 , Nc = CDp (∞)−1 , which leads to U = I. Note that Dp (∞)−Dp (s) is strictly proper, hence the term (s + a)(Dp (∞) − Dp (s)) is proper. Case ℓ∞ = 0 < ℓ = 1: In this case (2) becomes s−z Npz = I , z≥0, b>0, (6) s+b −1 and hence, P (s) = (s−z) (s+b) Dp Npr has one finite blockingzero at z ≥ 0. In the SISO case, these plants have relativedegree zero. The PIP is automatically satisfied because there is only one zero in C+e . Then the stable controller (s + b) † C= N (s) ( Dp (z) − Dp (s) ) (7) (s − z) pr strongly stabilizes P since (1) is satisfied with unimodular Dc = Dp (z)−1 , Nc = CDp (z)−1 , which leads to U = I. Case ℓ∞ = ℓ = 2: In this case (2) becomes 1 Npz = I , a1 > 0 , a 2 > 0 , (8) (s + a1 )(s + a2 ) 1 and hence, P = (s+a1 )(s+a Dp−1 Npr has two blocking2) zeros at infinity. In the SISO case, these plants have relative-degree two and are called minimum-phase. The PIP is automatically satisfied since there are no finite nonnegative zeros. Define D∞ (s) := (Dp (s)Dp (∞)−1 − I) which is strictly proper and stable. Choose any positive real β > 0 satisfying β > k s D∞ (s) k . (9) Let the controller be (s + a1 )(s + a2 ) † Npr (s) (Dp (∞) − Dp (s)). (10) C=β (s + β) Since Dp (∞) − Dp (s) is strictly-proper, the term (s+a1 )(s+a2 ) (Dp (∞) − Dp (s)) is proper; then all entries of (s+β) C are in RH∞ . With Dc = I, Nc = C, (1) becomes U = Dp (s) +
β (Dp (∞) − Dp (s)) (s + β)
1 sD∞ (s))Dp (∞). s+β For β satisfying (9), by the small-gain theorem, 1 k s+β sD∞ (s)k < 1, which implies U is unimodular. Therefore, the stable controller in (10) strongly stabilizes P . = (I +
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
Case ℓ∞ = 1 < ℓ = 2: In this case (2) becomes (s − z) Npz = I , z ≥ 0 , a > 0, b > 0 , (s + a)(s + b)
(11)
(s−z) Dp−1 Npr has a finite blockingand hence, P = (s+a)(s+b) zero at z ≥ 0 and one at infinity. In the SISO case, these plants have relative-degree one. Define W := Dp (z)−1 Dp (∞), which is obviously nonsingular. For these plants, the PIP is satisfied if and only if det W > 0 since det Dp is sign invariant at the blocking-zeros of P . The strongly stabilizing controller design proposed here assumes a more conservative condition on W : Let all eigenvalues of W = Dp (z)−1 Dp (∞) have positive real parts (which is equivalent to PIP for single-output plants since W > 0). Choose any real β > 0 satisfying (9) as in the case of 2-zeros at infinity above. With Npz as in (11), let the controller be −1 † Npr (s)(I − Dp (s)Dp (z)−1 )Dp (∞)(sI + βW )−1 , C = βNpz (12) (s+b) where (s−z) (I − Dp (s)Dp (z)−1 ) is stable. By assumption, −W has eigenvalues in the open left-half-plane and β > 0 implies (s + a)(sI + βW )−1 is also stable; then all entries of C are in RH∞ . With Dc = I, Nc = C, (1) becomes
U = Dp (s) + β(I − Dp (s)Dp (z)−1 )Dp (∞)(sI + βW )−1 = (sDp (s) + βDp (∞))(sI + βW )−1 1 sD∞ (s)) Dp (∞)(s + β)(sI + βW )−1 = (I + s+β 1 By (9) the term (I + s+β sD∞ (s)) is unimodular. Since Dp (∞) is nonsingular, and (s + β)(sI + βW )−1 is unimodular, the controller (12) is strongly stabilizing P .
The strongly stabilizing controller in (12) is designed under the condition that W = Dp (z)−1 Dp (∞) has eigenvalues in the open C+ ; while this is a sufficient condition for PIP for MIMO plants, it is equivalent to PIP for single-output systems. Case ℓ∞ = 0 < ℓ = 2: We only consider the case where (at least) one of the two finite blocking-zeros is at zero. In this case (2) becomes s (s − z) Npz = I , z ≥ 0 , b1 > 0 , b2 > 0 , (13) (s + b1 )(s + b2 ) s(s−z) Dp−1 Npr has a finite blockingand hence, P = (s+b 1 )(s+b2 ) zero at s = 0 and another at z ≥ 0. In the SISO case, these plants have relative-degree zero. Define V := Dp (0)−1 Dp (z), which is obviously nonsingular. For these plants, the PIP is satisfied if and only if det V > 0 since det Dp is sign invariant at the blocking-zeros of P . Similar to the previous case, the strongly stabilizing controller design proposed here assumes a more conservative condition on V : Let all eigenvalues of V = Dp (0)−1 Dp (z) have positive real parts (which is equivalent to PIP for single-output plants since V > 0). Let us define the stable transfer matrix Dp (s)Dp (0)−1 − I D0 (s) := s and choose any positive real ρ > 0 satisfying ρ k D0 (s) k < 1 . (14)
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Let the controller be (s + b1 )(s + b2 ) † C= Npr (s) [ Dp (z) − Dp (s) ] (sI + ρV )−1 , (s − z) (15) (s+b1 ) where (s−z) [ Dp (z1 ) − Dp (s) ] is stable. By assumption, −V has eigenvalues in the open left-half-plane and ρ > 0 implies (s + b2 )(sI + ρV )−1 is also stable; then all entries of C are in RH∞ . With Dc = I, Nc = C, (1) becomes U = Dp (s) + s(Dp (z) − Dp (s)) (sI + ρV )−1 = ( Dp (s)ρV + sDp (z) ) (sI + ρV )−1 = ((s + ρ − ρ)I + ρDp (s)Dp (0)−1 ) Dp (z)(sI + ρV )−1 s = (I + ρ D0 (s)) (s + ρ)Dp (z)(sI + ρV )−1 s+ρ s By (14) the term (I + s+ρ ρ D0 (s)) is unimodular. Since Dp (z) is nonsingular and (s + ρ)(sI + ρV )−1 unimodular the controller (15) is strongly stabilizing P . 4. CONNECTIONS WITH THE CONDITION OF ¨ ZEREN AND OZBAY In this section we show that a sufficient condition used to design strongly stabilizing controllers in Zeren and Ozbay (2000) is also necessary for the single-output version of the plants considered here. By (1), the plant P = Dp−1 Npz Npr is strongly stabilizable † if and only if there exists stable C = Npr Q, Q ∈ RH∞ , such that U := Dp + Npz Q is unimodular. Define Rp := Dp − I, then U = I + Rp + Npz Q . Using the small-gain theorem, a strongly stabilizing controller can be found if there exists Q ∈ RH∞ such that kRp + Npz Qk < 1. Since Npz (z) = 0 at all blocking-zeros z ∈ C+e , of P , a feasible Q ∈ RH∞ can be found only if kRp (z)k < 1. Now consider the plants described in Section 3 and let P have a single-output. Therefore Dp ∈ RH∞ and Npz ∈ RH∞ are scalars anr Npr has a single row. The only C+e zeros of the plant are in R+e , i.e., none are complex. By (2), a coprime factorization is P = Dp−1 Np = Dp−1 Npz Npr , where Npz contains the C+e zeros and Npr has a singlecolumn right-inverse in RH∞ . Without loss of generality, the denominator Dp ∈ RH∞ of P = Dp−1 Np can be 1 ⊂ R+ , p i ≥ expressed as follows: Let Pr = {pi }ki=1 0 denote the non-negative real poles and Pi = {ei ± 2 ⊂ C+ \ R+ , ei ≥ 0 denote the complex poles of jfi }ki=1 P in C+ . Then decompose Dp (s) as Dp (s) = Dpr (s)Dpc (s) (16) where k1 Y
Dpr (s) =
(s − pi ) (s + p i + ci ) i=1
Dpc (s) =
k2 Y
( (s − ei )2 + fi2 ) ( (s + ei )2 + fi2 + di s ) i=1
for some ci , di > 0 . Since Dp (∞) = 1, Rp = Dp − I is strictly-proper. Suppose that the plant is strongly stabilizable, equivalently the PIP holds (since Dp (∞) > 0, PIP implies
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
Dp (z) > 0). Suppose that the only C+e zeros of P are in R+e , i.e., P has no complex zeros in C+e . Let zmax ≤ ∞ ∈ R+e denote the largest zero of P . Suppose that pi ≤ zmax for all pi ∈ Pr , which always holds if P is strictly-proper. We now show that under these assumptions, |R(z)| < 1 at all R+e zeros of P : Let 0 ≤ z ≤ zmax be any of the R+e zeros of the plant, i.e., Npz (z) = 0. It is obvious that Q 2 ( (z−ei )2 +fi2 ) Dpc (z) = ki=1 satisfies 0 < Dpc (z) ≤ 1. ( (z+ei )2 +fi2 +di z ) For Dpr (z), there are two possibilities: Either this zero z > pi for all pi ∈ Pr , or z < pi for an even number of the poles pi ∈ Pr since the PIP implies an even number of positive real poles of P in the interval [z, zmax] (as well as between any two consecutive positive real zeros). In Qk1 (z−pi ) both cases, it is obvious that Dpr (z) = i=1 (z+pi +ci ) also satisfies 0 < Dpr (z) ≤ 1. Therefore, 0 < Dpr (z)Dpc (z) ≤ 1 implies |Rp (z)| = |1 − Dp (z)| < 1 at all R+e zeros of P . In the single-output case, the strongly stabilizing controller † can be written as C = Q = Npr Qp for some Qp ∈ RH∞ . Then using the sufficient condition of Zeren and Ozbay (2000), there exists a strongly stabilizing controller if inf ∞ kRp + Npz Qp k∞ < 1. (17) Qp ∈RH
For the single-output plant classes (4) and (8), condition 1 (17) is satisfied if and only if |Rp (∞)| < 1: For Npz = s+a , the optimal solution is Qp = (s + a)(Rp (∞) − Rp (s)) = (s + a)(Dp (∞) − Dp (s)); † † i.e., Q = Npr Qp is the same as controller C = Npr Ncz 1 in (5). For Npz = (s+a1 )(s+a2 ) , since Dp (∞) = 1 and β satisfies (9), the optimal solution is −1 −1 βNpz βNpz Qp = (Rp (∞) − Rp (s)) = (Dp (∞) − Dp (s)); s+β s+β † † i.e., Q = Npr Qp is the same as controller C = Npr Ncz in (10). For the cases (6) and (11) condition (17) is satisfied if and only if |Rp (z)| < 1. Necessity is obvious, for the sufficiency see e.g. Section 2.11 of the book by Foias et al (1996). For the case (6), Npz (s) = s−z s+b , the optimal solution is −1 −1 Qp = Npz (Rp (z) − Rp (s)) = Npz (Dp (z) − Dp (s)), † † i.e., Q = Npr Qp is the same as controller C = Npr Ncz in (7). In the case (13) the problem (17) is solvable if and only if |Rp (z)| < 1 and |Rp (0)| < 1. For systems with more than one zero in open C+ solvability condition for (17) is more complicated than just checking |Rp (zi )| < 1; but if one zero is in R+ and the other one is on the boundary of C+e , like in (13), then this statement is correct, see Section 2.11.3 of the book by Foias et al (1996).
We showed above that for (4), (8), (6) and (11) strong stabilizability implies |Rp (z)| < 1, which in turn implies condition (17). Therefore condition (17) is necessary and sufficient for these plants; i.e. there exist strongly stabilizing controllers (equivalently the PIP is satisfied) if and only if there is exists a stable Qp such that kRp + Npz Qp k < 1. Note that, here we do not require that the plant has no poles on the jω-axis, which is an assumption in Zeren and Ozbay (2000). Example. Consider the plant (18), which is taken from Zeren and Ozbay (2000),
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(s + 1)(s − 2) (s + 2)(s − 2) (18) (s2 + 4s + 5)(s − α) (s2 + 4s + 5)(s − α) with α ≥ 0. We see that this plant admits a factorization in the form P = Dp−1 Npz Npr , where P =
s−α s+1 s−2 Npz (s) = (s + a)(s + b) (s + a)(s + b) Npr (s) = 2 1 s + 4s + 5 Dp (s) =
s+2 . s+1
This fits into the form (11). Clearly Npz (2) = 0 and Npz (∞) = 0, so PIP is satisfied if and only if α < 2. The controller proposed in Section 3 for this type of systems is in the form (12), where we can take 1−c 2 s + 4s + 5 † (s + 1) Npr (s) = (s + a)(s + b) c (s + 2)
with any c ∈ R. We have W = Dp (2)−1 Dp (∞) = 3/(2−α), and W > 0 if and only if α < 2 (equivalent to PIP). Now we need to choose β > 0 such that β > ks D∞ (s)k. For this example, ks D∞ (s)k = (α + 1). Applying the formula (12) we obtain 1−c 2 −β(1 + α)(s + 4s + 5) (s + 1) , C= (19) 3β ) c (s + 2) (2 − α)(s + 1)(s + 2−α with any c ∈ R and any β > (α + 1). In this case we have U= =
s−α s+1
−
β(1 + α) (s − 2) 3β (2 − α) (s + 1)(s + (2−α) )
s2 + (β − α)s + β . 3β ) (s + 1)(s + (2−α)
Since β > α and α < 2, U is unimodular. Note that our condition β > ks D∞ (s)k gave β > (α + 1). But in fact β > α is sufficient. This gap is due to the use of the small gain theorem in our design of Section 3 in guaranteeing that U is unimodular. For the same system, we can obtain an alternative controller using the method of Zeren and Ozbay (2000). In that approach first note that (α + 1) Rp (s) = Dp (s) − 1 = − s+1 α+1 and |Rp (2)| = 3 < 1 if and only if α < 2 (equivalent to PIP). Solution of the problem (17) gives (α + 1)(s + b)(s + a) Qp (s) = − 3(s + 1)(1 + εs) for some sufficiently small ε > 0. In order to understand how small ε should be let us now examine U for the † controller C = Npr Qp , U (s) = =
s−α (α + 1)(s − 2) − s+1 3(s + 1)(1 + εs) 2(α+1) − εα εs2 + ( 2−α 3 − εα)s + 3 . (s + 1)(1 + εs)
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
This means that ε satisfying 0 < ε < 2−α 3α gives a strongly stabilizing controller 1−c (α + 1)(s2 + 4s + 5) (s + 1) . C(s) = − (20) c 3(s + 1)(1 + εs) (s + 2) Note that the two controllers (19) and (20) are identical if we set ε = 2−α 3β . 5. CONCLUSIONS In this paper we considered a class of MIMO plants with at most two blocking zeros in Re (no other blocking zeros in C+e ), and no restrictions were imposed on the number and locations of the poles. When they satisfy the PIP, these plants are shown to admit strongly stabilizing controllers and we have constructed such controllers explicitly. The order of the strongly stabilizing controllers developed here is equal to the order of the plant. We have also shown that for the single-output version of the class of plants considered here the sufficient condition of Zeren and Ozbay (2000), given for the construction of strongly stabilizing controllers, is equivalent to PIP. Strongly stabilizing controller design for MIMO plants with higher number of C+ blocking zeros is more difficult. For such plants one may have to check the existing sufficient conditions and design methods proposed earlier, see e.g. Campos-Delgado and Zhou (2003); Choi and Chung (2001); Chou et al. (2003, 2007); Gumussoy and Ozbay (2005); Lee and Soh (2002); Petersen (2006); Zeren and Ozbay (1999, 2000), and their references. For example, when we have two complex conjugate zeros placed in such a way that the PIP is about to be violated (as the imaginary part goes to zero) many of the existing finite dimensional (order of the controller is a few multiples of the plant order) controller design techniques fail. Because in this case the minimum order of the strongly stabilizing controllers can be very large (grows as the imaginary part gets smaller), see Smith and Sondergeld (1986). This is the main reason why we have not considered the two finite zero case. Our goal was to derive simple strongly stabilizing controllers for a class of plants using the PIP only. No additional conditions are imposed for this class of plants. REFERENCES Barabanov, A. E., “Design of H∞ optimal stable controller,” Proc. IEEE Conference on Decision and Control, pp. 734–738, 1996. Campos-Delgado, D. U., and K. Zhou, “A parametric optimization approach to H∞ and H2 strong stabilization,” Automatica, vol. 39, no. 7, pp. 1205–1211, 2003. Choi, Y., and W.K. Chung, “On the stable H∞ controller parameterization under sufficient condition,” IEEE Transactions on Automatic Control, vol.46, pp. 1618–1623, 2001. Chou, Y. S., T. Z. Wu, and J. L. Leu, “On strong stabilization and H∞ strong stabilization problems,” Proc. IEEE Conference on Decision and Control, pp. 5155–5160, 2003. Chou Y. S., J. L. Leu, and Y. C. Chu, “Stable Controller Design for MIMO Systems: An LMI Approach,” IET Control Theory Appl., Vol. 1, pp. 817–829, 2007.
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